Question: Simplify; express your answer in exponential form. Assume $a\neq 0, t\neq 0$. $\dfrac{{(a^{3}t^{-5})^{-2}}}{{(a^{-3}t^{-1})^{-4}}}$
Solution: To start, try simplifying the numerator and the denominator independently. In the numerator, we can use the distributive property of exponents. ${(a^{3}t^{-5})^{-2} = (a^{3})^{-2}(t^{-5})^{-2}}$ On the left, we have ${a^{3}}$ to the exponent ${-2}$ . Now ${3 \times -2 = -6}$ , so ${(a^{3})^{-2} = a^{-6}}$ Apply the ideas above to simplify the equation. $\dfrac{{(a^{3}t^{-5})^{-2}}}{{(a^{-3}t^{-1})^{-4}}} = \dfrac{{a^{-6}t^{10}}}{{a^{12}t^{4}}}$ Break up the equation by variable and simplify. $\dfrac{{a^{-6}t^{10}}}{{a^{12}t^{4}}} = \dfrac{{a^{-6}}}{{a^{12}}} \cdot \dfrac{{t^{10}}}{{t^{4}}} = a^{{-6} - {12}} \cdot t^{{10} - {4}} = a^{-18}t^{6}$